1.3 Dynamics

class: center, middle

# Engineering Physics - Fall 2024

---

<h3 style="color: dodgerblue;">Mechanics   <span style="color: black;font-size:50%;">$\mu\eta\chi\alpha\nu\iota\kappa\eta~\tau\epsilon\chi\nu\eta~=~$mode of action</span> </h3>

<p style="color: tomato;font-size: 50%">Kinematics   <span style="color: black;font-size:50%;">$\kappa\iota\nu\eta\mu\alpha~=~$motion</span>   
<span style="text-align: center;">description of motion disregarding <strong>forces.</strong></span>
</p>

<p style="color: tomato;">Dynamics   <span style="color: black;font-size:50%;">$\delta\upsilon\nu\alpha\mu\iota\zeta~=~$force</span>  
<span style="text-align: center;">motion of <strong>bodies</strong> under the action of <strong>forces.</strong></span>
</p>

<p style="color: black;font-size: 16px;">
As their <span style="color: dodgerblue;font-size:150%;">$1^{st}$ law</span>, Galileo and Isaac realized that if they (and no one else and nothing else) disturbs an apple, its <strong>velocity</strong> will not change:

\[
\frac{d}{dt}\vec{v}=0\quad.
\]

Seemingly more rebellious, Isaac ventured to quantify the effect of an external disturbance of the state of an object.
</p>

- <span style="color: gray;font-size:70%;">Clearly, the apple's velocity changed from when the wind detached it and the moment it fell into my lap.</span>
- <span style="color: gray;font-size:70%;">However, there must be more to the state of an apple than its velocity, as smaller/lighter apples did not fall.
$$\text{apple's state: inertia $\times$ velocity = momentum:}$$
$$\vec{p}=m~\vec{v}$$
</span>

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<p style="color: black;font-size: 16px;text-align: center;">
So <strong>Isaac</strong> proposed for the way the apple's state changes:
</p>

<div style="text-align: center;">
$
\vec{p}(t+\Delta t)=\vec{p}(t)+
$
<i class="fa fa-bolt" aria-hidden="true" style="color: tomato; font-size: 160%;"></i>
$
\cdot\Delta t
$
</div>

<p style="color: black;font-size: 16px;text-align: center;">
For the instantaneous change of state at time $t$:
</p>

<div style="text-align: center;">
$
\lim\limits_{\Delta t\to 0}\frac{\vec{p}(t+\Delta t)-\vec{p}(t)}{\Delta t}=\frac{d}{dt}\vec{p}=
$
<i class="fa fa-bolt" aria-hidden="true" style="color: tomato; font-size: 160%;"></i>
$
\equiv 
$
<a style="color: tomato;font-size: 120%;">$\vec{\text{Force}}$</a>
</div>

<p style="color: black;font-size: 16px;text-align: center;margin-top: 20px;">
<strong>Isaac</strong>: $\frac{d}{dt}m=0$, and my <span style="color: dodgerblue;font-size:150%;">$2^{nd}$ law</span> reads:
<div style="background: lightskyblue; padding: 1px;border-radius: 25px;width: 60%;margin: auto;">
\begin{eqnarray}
\frac{d}{dt}p_x=ma_x=m\frac{dv_x}{dt}=m\frac{d^2r_x}{dt^2}\equiv F_x\\
\frac{d}{dt}p_y=ma_y=m\frac{dv_y}{dt}=m\frac{d^2r_y}{dt^2}\equiv F_y\\
\frac{d}{dt}p_z=ma_z=m\frac{dv_z}{dt}=m\frac{d^2r_z}{dt^2}\equiv F_z
\end{eqnarray}
</div>
</p>

<p style="color: black;font-size: 16px;text-align: center;margin-top: 20px;">
Three equations which are succinctly written as one vector equation:
<div style="color: dodgerblue;font-size: 150%;">
\[
\vec{F}=m\vec{a}
\]
</div>
</p>

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<p style="color: black;font-size: 16px;text-align: center;margin-top: 20px;">
This equation marks ($\sim 1686$) the onset of a new era in science:
<div style="color: dodgerblue;font-size: 130%;">
    "Sum up all disturbances and you can predict the future!"
</div><div>
\[
\vec{F}_\text{friction}+\vec{F}_\text{gravity}+\vec{F}_\text{spring/Hooke}+\vec{F}_\text{electric}+\ldots=\sum_{\text{all forces}}\vec{f}=m\vec{a}
\]
</div>
</p>

<p style="color: black;font-size: 12px;text-align: center;margin-top: 20px;">
Visualization in a <a style="color:tomato;">free-body diagram:</a>
</p>

- <strong>one</strong> body
- attach all <strong>external</strong> force vectors acting on the body to <strong>center</strong> (of mass)
- include coordinate system

<div class="responsive">
  <div class="gallery">
    <a target="_blank" href="/images/FBD_1.png">
      <img src="/images/FBD_1.png" alt="Cinque Terre" width="600" height="400">
    </a>
    <div class="desc">Draw separate FBD's including gravitational and normal forces even if the cancel one another.</div>
  </div>
</div>

<div class="responsive">
  <div class="gallery">
    <a target="_blank" href="/images/FBD_2.png">
      <img src="/images/FBD_2.png" alt="Forest" width="600" height="400">
    </a>
    <div class="desc">Find the tension.</div>
  </div>
</div>

<div class="responsive">
  <div class="gallery">
    <a target="_blank" href="/images/FBD_3.png">
      <img src="/images/FBD_3.png" alt="Northern Lights" width="600" height="400">
    </a>
    <div class="desc">Assume that the system does not move. Express the angles as functions of $m_{1,2,3}$.</div>
  </div>
</div>

<div class="responsive">
  <div class="gallery">
    <a target="_blank" href="/images/FBD_4.png">
      <img src="/images/FBD_4.png" alt="Mountains" width="600" height="400">
    </a>
    <div class="desc">Assume that a static, frictional force $f$ keeps the block stationary until the angle $\theta$ reaches a critical value.
    What is this value $\theta_c$ if you assume that below that, $f$ always equals the downward pull?</div>
  </div>
</div>

<p style="color: tomato;font-size: 12px;text-align: center;margin-top: 20px;">
Pulleys, ropes are assumed massless, and ropes glide frictionless on pulleys. At each point <strong>inside</strong> a rope
the tension in opposite directions is equal. At the connection points to an object, the ropes exert the force of tension on
those objects.
</p>

---

<div class="grid-item">
<span style="text-align: center;font-size: 16px;">
<strong>Idea:</strong> Understand the discrete form of \[m\frac{d^2\vec{r}(t)}{dt^2}=\vec{F}(x)\] as a <strong>recurrence relation</strong> for position and velocity.
</span>
<br>
<ul  style="list-style-type:square;text-align: left;font-size: 16px;">
<li style="margin-bottom: 10px;">Represent the disturbing force as a function of position, time, and velocity<br>
  <div style="font-size:10px;color:tomato;">e.g.:  <img src="images/mass.gif" width=200>   $F(x)=k_1(x_1-x)+k_2(x-x_2)$</div>
<li style="margin-bottom: 10px;">Set   $x_0$   and   $v_0$   (initial conditions)</li>
<li style="margin-bottom: 10px;">Repeat the following block for   $t=0,\,\Delta,\,2\Delta,\,\ldots:$</li>

<div style="border: 1.5px solid darkred;margin-top: 5px;margin-bottom: 5px; padding-right: 5px;padding-top: 2px;padding-bottom: 3px;">
<ol style="font-size: 12px;">
<li style="margin-bottom: 5px;"> Update the position:  
$
x_{t+\Delta}=x_t+\Delta\,v_{t'}
$
 with earlier velocity and position<br>
<div style="font-size:10px;color:dodgerblue;">
  (explore the different options:  $t'=t+\kappa\cdot\Delta$   with   $\kappa\in[0,1]$)<br>
</div>
</li>
<li style="margin-bottom: 0px;"> Update the velocity:
$
v_{t'}=v_{t'-\Delta}+\frac{\Delta}{m}\,F(x_t)
$
<br>
with   $v_{\kappa\Delta}=v_{0}+\frac{\kappa\Delta}{m}\,F(x_t)$   for the special case   $t=0$
</li>
</ol>
</div>
</ul>
<br>
<a href="./1dsimul.html" style="display: inline-block;text-align: center;background-color: gray;color: white;padding: 8px 10px;border-radius: 25px;"> Interactive Python lab <img src="./images/python.svg" width=26 height=26>
</i></a>
<p style="color: black;font-size: 12px;text-align: center;">
<strong>Usage:</strong> modify to your likes and then click  <i class="fa fa-caret-right" style="color:green;font-size: 220%;"></i>
</p>
</div>
</div>